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Hindawi Publishing CorporationInternational Journal of Quality, Statistics, and ReliabilityVolume 2011, Article ID 718901, 10 pagesdoi:10.1155/2011/718901

Research Article

Probabilistic Modeling of Fatigue Damage Accumulation forReliability Prediction

Vijay Rathod,1 Om Prakash Yadav,2 Ajay Rathore,1 and Rakesh Jain1

1 Department of Mechanical Engineering, Malaviya National Institute of Technology, Jaipur 302017, India2 Department of Industrial and Manufacturing Engineering, North Dakota State University, Fargo, ND 58108, USA

Correspondence should be addressed to Om Prakash Yadav, [email protected]

Received 2 December 2010; Revised 15 March 2011; Accepted 16 March 2011

Academic Editor: Ajit K. Verma

Copyright © 2011 Vijay Rathod et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A methodology for probabilistic modeling of fatigue damage accumulation for single stress level and multistress level loadingis proposed in this paper. The methodology uses linear damage accumulation model of Palmgren-Miner, a probabilistic S-Ncurve, and an approach for a one-to-one transformation of probability density functions to achieve the objective. The damageaccumulation is modeled as a nonstationary process as both the expected damage accumulation and its variability change withtime. The proposed methodology is then used for reliability prediction under single stress level and multistress level loading,utilizing dynamic statistical model of cumulative fatigue damage. The reliability prediction under both types of loading isdemonstrated with examples.

1. Introduction

Most of the mechanical components are subjected to fatiguedue to random loading as well as constant amplitude loadingduring their usage. Fatigue is also recognized as one of themain reasons for failure of mechanical components [1]. Thishas led to a need for developing new approaches to predictthe reliability and useful life of mechanical components,which are subjected to fatigue damage. This has been theprimary focus of designers for many years, and the field stillpresents many challenges, even though extensive progress hasbeen made in the past few decades [2].

Earlier models of fatigue damage accumulation reportedin the literature focus on deterministic nature of the processwhereas in practice, damage accumulation is of stochasticnature. This stochasticity results from the randomness in fa-tigue resistance of material as well as that of the loading proc-ess [3]. As a result of this, even under constant amplitudefatigue test at any given stress level, the fatigue life shows sto-chastic behavior with a specific distribution. The literatureshows that fatigue life data follows either normal or log-normal distribution under constant amplitude or randomloading [4–6]. Weibull distribution has also been reported to

fit fatigue life data [7, 8], though there are no apparent phys-ical or mathematical phenomenon explained for this [9].Researchers have proposed different modeling approaches tothe probabilistic damage accumulation paradigm. Shen et al.[3] developed a probabilistic distribution model of stochasticfatigue damage, wherein they have considered the random-ness of loading process as well as the randomness of fatigueresistance of material by introducing a random variableof single cycle fatigue damage. Liu and Mahadevan [2]proposed a general methodology for stochastic fatigue lifeunder variable amplitude loading by combining a nonlinearfatigue damage accumulation rule and stochastic S-N curverepresentation technique. Nagode and Fajdiga [10] havemodeled a probability density function (PDF) of failurecycles at any stress level as a normal distribution based on theDeMoivre-Laplace principle to reliably predict endurancelimit of a randomly loaded structural component. Liao et al.[11] proposed a new cumulative fatigue damage dynamicinterference model assuming that cumulative damage followsnormal or lognormal distribution. Wu and Huang [12]modeled fatigue damage and fatigue life of structural com-ponents subjected to variable loading as a Gaussian randomprocess. Ben-Amoz [13] developed a cumulative damage

2 International Journal of Quality, Statistics, and Reliability

theory based on the concept of bounds on residual fatigue lifein two-stage cycling. Castillo et al. [14] developed a generalmodel for predicting fatigue behavior for any stress level andrange by generalizing the Weibull model. Sethuraman andYoung [15] have developed a cumulative damage thresholdcrossing model. This model considers a multicomponentproduct which undergoes deterioration/damage at regularintervals of time and failure occurs as soon as the maximumdamage to some component crosses a certain threshold.Time to failure data is used to estimate the model parameters.A comprehensive review of cumulative fatigue damage andlife prediction theories can be found in Fatemi and Yang [16].

Two aspects are significantly important from the pointof view of modeling probabilistic fatigue damage. First, anaccurate physical damage accumulation model needs to bein place to predict expected or nominal fatigue damage.Second, an appropriate uncertainty modeling techniqueis required to account for stochasticity [2]. A review ofliterature has indicated that handling of stochasticity inmodeling uncertainty involve complex mathematics. Thisfact is the primary motivation behind the development of asimpler approach for handling stochasticity in fatigue dam-age accumulation modeling in the proposed research work.This paper proposes a simpler approach to deduce the distri-bution of a fatigue damage accumulation from the fatiguelife distribution using a one-to-one transformation meth-odology and to a certain extent attempts to minimize themathematical complexity. It also proposes a simple andunique way to model the damage accumulation processtreating it as a nonstationary probabilistic process to capturedamage accumulation and its variability at any given pointof time. The proposed methodology can be effectively usedto predict reliability of mechanical components subjectedto fatigue loading due to single stress level and multi-stresslevel.

An outline of this paper is as follows. Section 2 elaboratesan approach for modeling probabilistic damage accumula-tion to capture expected values of damage and its variability.Section 3 presents a reliability prediction approach utilizingdynamic model of cumulative fatigue damage. The appli-cability of proposed methodology and its implications areillustrated with help of examples in Section 4. The conclu-sion is presented in Section 5.

2. Modeling Probabilistic FatigueDamage Accumulation

Damage accumulation is a complex and irreversible phe-nomenon, wherein the damage of the product underconsideration gradually accumulates and over a period oftime leads to its failure. Therefore, damage accumulationcan be treated as a measure of degradation in fatigueresistance of materials. Moreover, the damage accumulationis probabilistic in nature and it can be depicted graphicallyas shown in Figure 1. The Figure shows monotonicallyincreasing degradation path where the degradation measureis increasing probabilistically with time.

Failure threshold

Time, t

Deg

rada

tion

mea

sure

Figure 1: Degrading path example.

Wang and Coit [17] have explained that at any specifiedtime, there exists a distribution of degradation measure-ments considering a population of similarly degradingcomponents. They also pointed out that the variability inany given degradation measure increases with usage time.Since damage accumulation is also a measure of degradation,the reasoning given by Wang and Coit [17] can be appliedin assuming that damage accumulation also follows certainprobability distribution and that the expected value andvariability of any damage accumulation measure will increasewith usage time. Further, Wu et al. [6] have shown thatunder constant or random amplitude loading conditionsnormal or lognormal distribution provides good fit to fatiguefailure data. Therefore, in the proposed work the damageaccumulation is modeled as a nonstationary probabilisticprocess assuming that fatigue life follows normal distribu-tion. Furthermore, since damage accumulation is a functionof uses cycle, the probability distribution of damage accu-mulation can be treated as normal distribution as advocatedin Benjamin and Cornell [18]. The nonstationary Guassianprocess of damage accumulation can be given as:

D(t) ≈ N{μD(t), σ2

D(t)}

, (1)

where D(t) is a damage accumulation measure that variesprobabilistically with time t, and μD(t) and σ2

D(t) are itsmean and variance. The proposed probabilistic modeling ofdamage accumulation is elaborated in subsequent sections.

2.1. Modeling Expected Value of Damage Accumulation. Thetwo most widely used models for fatigue loading are S-Ncurve and Palmgren-Miner’s damage accumulation models[2, 19]. The S-N curve model is used to express therelationship between fatigue life (Nf ) and stress level (S) andis expressed by the well-known S-N curve equation as givenbelow:

Nf Sm = A, (2)

where A is a fatigue strength constant and m representsslope of the S-N curve. Figure 2 shows a probabilisticinterpretation of the S-N curve, wherein PDFs (on normal

International Journal of Quality, Statistics, and Reliability 3

PDFs of fatigue life

Nf 3Nf 2Nf 1 Number of cycles n

S1

S2

S3

Stre

ssS N f Sm = A

Figure 2: Probabilistic S-N curve.

scales) of fatigue lives are depicted at different stress levels S1,S2 , and S3.

The linear damage accumulation model, which is alsoknown as Palmgren-Miner’s rule, defines damage as the ratioof the number of cycles of operation to the number of cyclesto failure at any given stress level [19]. Assuming no initialdamage, the damage accumulation at single stress level isgiven as:

D = n

Nf. (3)

Similarly, for multi-stress levels, damage accumulation canbe expressed as:

D =k∑

i=1

Di =k∑

i=1

niN f i

, (4)

where D is the total accumulated fatigue damage of thematerial, Di is the damage accumulated when subjected toith stress level, ni is the number of usage cycles, and Nf i isthe number of cycles to failure at the ith stress level. It isassumed that failure occurs when total damage accumulationreaches unity [6, 19]. The number of cycles to failure (Nf i)at any given stress level can be obtained from the S-N curvemodel. Therefore, by considering both S-N curve model andthe linear damage accumulation model, a linear relationshipmodel between damage accumulation and number of usagecycle at any given stress level can be derived by combining (2)and (3) as given below:

D = Sm

An,

D = CSmn,

(5)

where C represents a reciprocal of fatigue strength constant,S denotes a constant amplitude stress level, m representsslope of the S-N curve, and n denotes number of usagecycles. Similarly, for multi-stress levels the linear damageaccumulation model can be derived by modifying (5) asfollows:

D =k∑

i=1

Di =k∑

i=1

CSmi ni. (6)

Transformed/derivedPDF of damage

accumulationat failure life

Known PDF of fatigue life

at stress level S

m′1

l1

O

Equal shaded areas

fd(D)dD = fn(Nf )dN f

N f Number of usage cycles n0

D = 1

Dam

ageD

Figure 3: One-to-one transformation of PDF.

Equations (5) and (6) provide expected value of damage ac-cumulation at any given point of time (usage cycles) subject-ed to single stress level and multiple-stress levels, respectively.However, in order to come up with a realistic estimate ofreliability of any given product, it is necessary to adopt aprobabilistic approach. It is, therefore, important to treatthe damage accumulation measure under consideration asa random variable and to establish an appropriate distribu-tion of damage accumulation. In the following sections, amethodology is proposed to derive the PDF of the damageaccumulation measure by considering the PDF of fatigue life(Nf ).

2.2. Distribution of Fatigue Damage Accumulation. In orderto establish the PDF of the damage accumulation measure(D) and to estimate distribution parameters, first the fatiguefailure life is treated as random variable which follows certaindistribution. Thereafter, the distribution of D is derivedusing the one-to-one PDF transformation methodologyproposed by Benjamin and Cornell [18]. As per Benjaminand Cornell [18], the unknown PDF of a random variablecan be derived using this transformation technique, if thatvariable is directly or functionally related to another randomvariable whose PDF is already known. Since cumulativedamage accumulation is a function of usage life (or fatiguefailure life), the PDF transformation methodology providesan effective means to establish distribution of damage accu-mulation measure. In Figure 3, line l1 is the trend lineof expected damage accumulation as given by (5) at agiven stress level (S). The straight line depicts a linearrelation between the damage accumulation measure andusage cycles (n). Now assuming that the PDF of usage cyclesis known, Figure 3 provides the basic understanding ofhow this known PDF of fatigue life can be used to obtainthe PDF of the damage accumulation measure. It must benoted, as shown in Figure 3, that initial variability of usagecycles is zero and it follows increasing trend in variabilitywith increase in usage cycle. It is further assumed that thisincrease in variability also follows a linear trend as suggestedby Coit et al. [20].

From the above discussion, it is clear that in order toderive the distribution of D using Benjamin and Cornell’s[18] PDF transformation technique, there are two basic


requirements that need to be fulfilled: (i) a clearly definedrelation between damage accumulation and usage cycles and(ii) the knowledge of the distribution or PDF of the usagecycle.

In order to satisfy the first requirement, a linear relation-ship between the damage accumulation measure and usagetime for a given stress level can be established by redefining(5) as follows:

D = m′n, (7)

where m′ represents the slope of the damage accumulationtrend line. At fatigue failure life (i.e., at n = Nf ), (7) can bewritten as:

D = m′Nf . (8)

As mentioned earlier, under constant or random amplitudeloading conditions, normal or lognormal distributions pro-vide good fit to fatigue failure data than Weibull distribution[6]. Therefore, in the proposed work, it is assumed thatthe fatigue failure life Nf follows normal distribution. Thisfulfills the second requirement of known distribution offatigue failure life too. The PDF of normally distributedfatigue life is given as:

fn(Nf

)= 1σNf

√2π

exp

⎛

⎝−12

(Nf − μNf

σNf

)2⎞

⎠. (9)

2.2.1. Application of the PDF Transformation Technique forDeriving the PDF of the Damage Accumulation Measure.The functional relationship between damage accumulationmeasure and fatigue life (Nf ) can be generically expressed asgiven below:

D = g(Nf

). (10)

The inverse relation of (10) can be expressed as:

Nf = g−1(D). (11)

The cumulative distribution function (CDF) of the depen-dent variable D can be obtained from the CDF of Nf as:

Fd(D) = Fn(g−1(D)

). (12)

Subsequently to obtain the PDF of the damage accumulationmeasure (D), take a derivative of its CDF as given below [18]:

fd(D) = d

dDFd(D)

= d

dDFn(g−1(D)

)

= d

dD

[∫ g−1(D)

−∞fn(Nf

)dNf

]

= dg−1(D)dD

fn(g−1(D)

).

(13)

Using (11), (13) can be written in a more suggestive form asfollows:

fd(D) = dNf

dDfn(Nf

)(14)

or

fd(D)dD = fn(Nf

)dNf . (15)

The graphical interpretation of (15) is shown in Figure 3by equal shaded areas. As shown graphically, the intervalwidths dD and dNf are not equal, but in case of thelinear relationship between random variables, the ratio ofdNf /dD is constant [18].

Further, differentiating (8) with respect to D gives:

dNf

dD= 1m′ . (16)

Substituting (16) and (9) in (14) gives the PDF of D asfollows:

fd(D) = 1m′σNf

√2π

exp

⎛

⎝−12

((D/m′)− μNf

σNf

)2⎞

⎠. (17)

From (17), it is clear that the damage accumulation also fol-lows similar distribution, and the relation between standarddeviations before and after transformation is as follows:

σD = m′σNf , (18)

where σD represents standard deviation of damage accumu-lation, σNf denotes standard deviation of failure life or usagecycle, and m′ represents slope of damage accumulation trendline. This clearly indicates that variability in damage accumu-lation depends on slope of the damage accumulation trendline and variability in usage cycle.

Assuming that slope of the damage accumulation trendline is constant for any given stress level, the variability indamage accumulation can be treated as a function of thevariability in usage cycle. Equations (17) and (18) derivedusing one-to-one transformation approach clearly explainthat the PDF of damage accumulation will be the same asusage life distribution because of linear relationship betweenthese two variables. However, the resulting PDF of damageaccumulation is scaled down by factor m′.

2.3. Modeling Trend Line of Variance. As already discussed,the damage accumulation increases linearly with usage cyclesat a given stress level. Further, Figure 4 shows increasingtrend in variability or standard deviation of fatigue lives withdecrease in stress levels as well. It must also be noted thatthe variability is zero at the initial stage (i.e., at n = 0) forall the stress levels. This indicates the presence of a trend ofincreasing variance in fatigue lives as stress level decreases,that is, there is low variability in fatigue life at higher stresslevel and higher variability in fatigue life when product issubjected to lower stress levels as shown in Figure 4. Thechallenge is to capture this increasing trend in variability with


PDFs of fatigue livesat stress levels S1, S2, and S3

(S1 > S2 > S3)

Nf 3Nf 2Nf 1

Number ofusage cycles n

0

D = 1

Dam

ageD l1 l2 l3

Figure 4: Damage accumulation for different stress.

stress levels as well as usage cycle and model it into totalvariability of damage accumulation.

Based on the above discussion, there are two interestingaspects to be considered for modeling total variability indamage accumulation.

(1) At constant stress level, the variability in damageaccumulation increases monotonically with increasein usage cycles as explained by Wang and Coit [17]and Coit et al. [20].

(2) The variability in fatigue life is lower at higherstress levels, whereas it monotonically increases withdecrease in stress levels as shown in Figures 2 and 4.Pascual and Meeker [21] have also stated that inmost fatigue experiments, the variance of the fatiguelife increases as stress decreases, and therefore thestandard deviation of fatigue life can be modeled asmonotonic function of stress. In fact, the variabilityin fatigue life of a product subjected to multiple stresslevels is another major source of variability in damageaccumulation.

In order to model total variability in damage accumulation,these two sources of variability need to be considered. Thefollowing section elaborates the modeling of variability indamage accumulation considering these two aspects.

2.3.1. Deriving the Equation for the Variance Trend Line.Given the fact that the variability in usage cycles continuouslyincreases from zero to a certain value at fatigue failure life, itsrate of change can be interpreted using geometric reasoning.This is shown diagrammatically in Figure 5.

As shown in Figure 5, l1 represents the mean damageaccumulation trend line and line l2 connects the origin with1σ point (i.e., point a in Figure 5) of the fatigue life dis-tribution. Therefore, ao′ represents one standard deviationdistance from center point of the fatigue life distribution, andhence, σNf = ao′. Let us assume that slopes of these two linesl1 and l2 are represented as m′ and m′′, respectively.

PDF of fatigue lifeat stress S

n2 = Nfn1Number of

usage cycles n

o

D = 1

Dam

ageD

a b o′

dce

l2l1

f g

Figure 5: Geometric interpretation of rate of change of variabilityof usage cycles.

From the geometric construction shown in Figure 5, therate of change of standard deviation (rσ ) of usage cycles canbe interpreted as:

rσ =(ao′ − ce)(n2 − n1)

, (19)

where ao′ = ab + bo′, ce = cd + de, and bo′ = n2 − n1.Substituting these in (19) gives:

rσ =(ab + bo′ − cd − de)

(bo′). (20)

Since ab = de, we have:

rσ =(

1− (cd)(bo′)

)

. (21)

The slope of mean damage accumulation line l1 can beobtained as:

m′ = be

bo′. (22)

Similarly, the slope of line l2 can be derived as:

m′′ = ad

cd. (23)

By taking the ratio of two slopes and considering that ad =be, it gives:

cd

bo′=(m′

m′′

). (24)

Substituting (24) in (21) gives:

rσ =(

1− m′

m′′

). (25)

Further, the slope of lines l1 and l2 can also be obtained as:

m′ = o′gog

= 1Nf

,

m′′ = a f

o f= 1Nf − σNf

.

(26)


Using (26) in (25) gives:

rσ =(

1−(Nf − σNf

N f

))

,

rσ =σNf

N f.

(27)

The variability (or standard deviation) in usage cycle (n) atconstant amplitude stress level (S) can be given as:

σn = rσn. (28)

Now by combining (27) and (28), the standard deviation ofusage cycle is estimated as:

σn =(σNf

N f

)

n. (29)

Considering (18) and (29), a relationship can be derivedto estimate variability or standard deviation of damageaccumulation measure (D) as follows:

σD =(σNf

N f

)

nm′. (30)

Considering m′ = CSm, (30) can be modified in a moresuggestive form as follows:

σD = CSmn

(σNf

N f

)

. (31)

Equation (31) indicates that variability in damage accu-mulation is a function of stress amplitude, usage cycles,fatigue failure life and variability in fatigue failure life. Inorder to estimate variability in damage accumulation, theseparameter values can be obtained from the probabilistic S-Ncurve (refer to Figure 2).

The above equation represents a model that can be usedto capture the variability in damage accumulation for a singlestress level. The same model can be extended to capturethe variability in damage accumulation if the product issubjected to multi-stress level as well. Figure 6 illustratesthe concept of how variance of usage cycles changes in amultilevel stress loading scenario.

It has already been discussed (refer to Figure 4) that therate of damage accumulation and its variability depend onstress levels and number of usage cycles. Moreover, the rateof change of variability is different for different stress levels.Therefore, considering multi-stress level scenario, the totalvariability or standard deviation in damage accumulation atthe time of fatigue failure can be obtained using the followingequation:

σD =

√√√√√

k∑

i=1

(

CSmi ni

(σNf i

N f i

))2

, (32)

where suffix i represents the level of stress in the multilevelstress loading scenario.

In the next section, the models developed for capturingdamage accumulation and its variability are used in conjunc-tion with the dynamic statistical model for predicting thereliability of mechanical components.

Nfn3n2n1

O-O1: Damage accumulation at stress level S1

O1-O2: Damage accumulation at stress level S2

O2-O3: Damage accumulation at stress level S3

O1

O2

O3

Number ofusage cycles n

Transformed/derived PDF of damage

O

D = 1

Dam

ageD

Figure 6: Damage accumulation for multi-stress loading and itsdistribution.

3. Reliability Prediction

The well-known stress-strength interference model considersproduct reliability from the probabilistic point of view. Thisconcept has been used by many researchers for developingmodels to predict product reliability in the past [11, 22–24].Liao et al. [11] have classified existing cumulative fatiguedamage models for reliability prediction into two groupsbased on fundamental assumptions and hypothesis as: (i)static statistical models and (ii) dynamic statistical models.Unlike static models, dynamic models treat both expectedvalue and variance of random variable as time dependent andtheir values continuously change with time. However, thesedynamic statistical models are developed on existing classicalstress-strength interference model considering random vari-able as dynamic random variable [11]. In the present work,the damage accumulation is treated as dynamic randomvariable whose distribution parameters (mean and variance)are dependent on usage life (time) as given by (5) and (31).This paper proposes a dynamic reliability prediction modelconsidering probabilistic damage accumulation developed inthe previous section of this paper. The following assumptionshave been made while formulating a dynamic reliabilityprediction model.

(1) Fatigue failure occurs when damage accumula-tion (D) reaches the threshold damage (Dc), whereE(Dc) = 1.

(2) The threshold damage or critical damage has thesame distribution as the damage accumulation mea-sure.

(3) When usage life is equal to the fatigue failure life (n =Nf ), the variability of threshold damage accumula-tion (σ2

Dc) is equal to the variability of damage accu-

mulation measure (σ2D). The variability of damage

accumulation measure continuously increases withusage life but when usage cycle reaches to fatiguefailure level, the corresponding variability is assumed


to be the same as variability of threshold damageaccumulation. However, it is statistically independentof (D).

Since the damage accumulation measure is treated as anormally distributed dynamic variate, the reliability of aproduct in terms of damage accumulation can be modeledas follows:

R = prob(D < Dc)

= 1− prob(Dc −D ≤ 0)

= 1−∅⎛

⎝−(μDc − μD

)

√σ2Dc

+ σ2D

⎞

⎠.

(33)

A diagrammatic representation of the above conceptis shown in Figure 7. It is important to note that whenusage cycle is equal to failure life (n = Nf ), the variabilityof threshold damage accumulation will be equal to thevariability of damage accumulation (σ2

Dc= σ2

D).Substituting (6) and (32) in (33), the reliability can be

expressed in a more suggestive form as follows:

R = 1−∅

⎛

⎜⎜⎝−

(μDc −

∑ki=1 CS

mi ni

)

√

σ2Dc

+∑k

i=1

(CSmi ni

(σNf i /N f i

))2

⎞

⎟⎟⎠.

(34)

The above model provides a dynamic reliability predictionconsidering dynamic behavior or continuous degradationphenomenon of the product with usage cycle. In essence,the proposed dynamic reliability prediction model capturesproduct life cycle and assesses product reliability for a giventime period or usage cycle. The proposed model can be usedfor predicting reliability of a product subjected to both singlestress and multi-stress level scenarios. The applicability of theproposed model is demonstrated with the help of a helicalspring (compression type) example.

4. Example

To demonstrate the applicability of the proposed reliabilityassessment approach, fatigue test data required to model S-N curve is adopted from Zaccone [7], which were obtainedafter conducting fatigue tests on helical compression springsused in compressors. Table 1 shows the fatigue test data atdifferent amplitude stress levels and corresponding standarddeviation considered.

Using the above data, the S-N curve model was fitted toobtain the model parameters as follows:

m = 8.46; A = 1.3544× 1027. (35)

The value of constant C is calculated by taking reciprocal ofthe fatigue strength constant and is obtained as:

C = 7.3829× 10−28. (36)

Using these parameters values and (6), the expected value ofdamage accumulation in the compression spring for single

Nonstationary PDF of damage

f (μD) f (μDc )

μDcμD

f

Figure 7: Dynamic stress-strength interference model for damageaccumulation.

Table 1: Fatigue test data.

Constant amplitudestress (MPa) (Si)

Mean number of cycle tofailure (Nf )

Standarddeviation (σNf )

470 33581 4785

435 64616 9692

395 146101 23291

360 320222 53947

320 867130 156303

stress or multi-stress level for given number of usage cyclescan be obtained as follows:

μD =k∑

i=1

7.3829× 10−28 × S8.46i × ni. (37)

4.1. Reliability Prediction for a Single Stress Level. First, theapplicability of the proposed model is demonstrated byestimating the reliability for single stress level. A single stresslevel of 360 MPa is considered. For that purpose, one hasto estimate the variability of the threshold damage (σDc) atfatigue failure life and the variability of damage accumula-tion at any given usage cycle. The variability of thresholddamage at fatigue failure life is calculated considering thirdassumption (σDc = σD) at the fatigue failure life (Nf ) andusing (31) as follows:

σDc = CSmn

(σNf

N f

)

, (38)

where C = 7.3829 × 10−28, S = 360 MPa, fatigue failure lifen = Nf = 320222 cycles, and σNf = 53947 cycles.

Using these values of variables and parameters, thevariability at threshold damage is obtained as (σDc) =0.16846.

Similarly, one can calculate the variability in damageaccumulation at any given time period or usage cycle (n).Once this element of variability is estimated, (34) can be usedto estimate reliability of the compression spring subjected tosingle stress level for any given time period as given below:

R = 1−∅

⎛

⎜⎜⎝−

(1−H)√

0.168462 +(H ×

(σNf /N f

))2

⎞

⎟⎟⎠, (39)


×105987654321

Number of usage cycles, n

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Rel

iabi

lity,R

Reliability at 435 MPa



Figure 8: Reliability plot at single stress levels of 435, 360 and320 MPa.

where (H) denotes (7.3829 × 10−28 × S8.46 × n). Equation(39) expresses the reliability as a function of usage cycles (n)at single stress level.

Considering reliability function equation (39), Figure 8illustrates a reliability plot for single stress level at 435 MPa,360 MPa, and 320 MPa. These reliability plots clearly revealthe trend of reliability loss with increase in usage cycles. Thecareful analysis of these reliability plots indicate that reliabil-ity remains higher (almost constant) for initial period andlater on starts declining with usage cycle. This phenomenonexplains existing understanding of crack initiation andcrack propagation periods. The higher and stable reliabilityphase, although it varies with stress levels, represents crackinitiation period, and reliability loss phase is indicative ofcrack propagation period. It is clear from these plots thatcrack initiation period is smaller for higher stress level andloss of reliability is faster during crack propagation indicatingfaster degradation or higher rate of damage accumulation.The total life of the product also varies with stress levels andtherefore supports out argument that dynamic behavior ordegradation phenomenon needs to be captured in designoptimization models to provide more realistic solutions.

4.2. Reliability Prediction for Multi-Stress Levels. To demon-strate multi-stress loading scenarios, let’s consider a helicalcompression spring subjected to three successive stress levelsof 435 MPa, 360 MPa, and 320 MPa for 40000, 60000 andremaining number of usages cycles up to fatigue failure,respectively. To estimate reliability under multi-stress levelloading, the fatigue life of spring needs to be predicted undermulti-stress loading conditions.

4.2.1. Prediction of Fatigue Life. As mentioned earlier, thelinear damage accumulation theory states that the damagefraction at any stress level is linearly proportional to the ratioof the number of usage cycles to the number of cycles to

fatigue failure at that stress level [19]. In the case of multi-stress level loading, when these damage fractions equal unity,fatigue life can be predicted. Therefore, to predict fatiguelife first, the remaining life (n3) of the spring after firstand second stress levels needs to be calculated. This can beobtained as follows:

n3 = 1CS8.46

3

⎛

⎝1−2∑

i=1

CS8.46i ni

⎞

⎠. (40)

The remaining life is obtained as n3 = 167750 number ofusage cycles.

The estimated fatigue life (Nf e) of spring under multi-stress level loading condition is given as:

Nf e = (40000 + 60000 + 167750) = 267750, (41)

number of usage cycles.

4.2.2. Prediction of Reliability. To estimate the reliabilityof spring subjected to given multilevel stress loading, thevariability of the threshold damage at fatigue failure lifeneeds to be estimated. The variability of the thresholddamage at fatigue failure life is calculated as given inSection 4.1 (i.e., using third assumption) and(32)

σDc =

√√√√√

3∑

i=1

(

7.3829× 10−28 × S8.46i ×ni ×

(σNf i

N f i

))2

,

(42)

where usage cycles are n1 = 40000, n2 = 60000, and n3 =167750, and fatigue failure livesNf 1 = 64616,Nf 2 = 320222,Nf 3 = 867130 , and variability at each fatigue failure lives areσNf 1 = 9692, σNf 2 = 53947, and σNf 3 = 156303 .

Using the appropriate values of variables and parameters,the variability at the threshold damage (σDc) is σDc =0.16042.

Similarly, the variability in damage accumulation at anygiven usage cycle can be obtained. After estimating thevariability at threshold damage and variability at any givenusage cycle, the reliability of the compression spring for anygiven time period subjected to given multilevel stress loadingcan be estimated using equation given below:

R = 1−∅

⎛

⎜⎜⎝−

(1−∑3

i=1 K)

√

0.168462 +∑3

i=1

(K ×

(σNf i /N f i

))2

⎞

⎟⎟⎠,

(43)

where (Ki) denotes (7.3829 × 10−28 × S8.46i × ni). Using

reliability function equation (43), a reliability plot formulti-stress level loading of three successive stress levels of435 MPa, 360 MPa, and 320 MPa is illustrated as plot-A inFigure 9. A careful analysis of reliability plot-A indicates thatinitially the damage accumulation rate is higher because thespring is first subjected to higher stress level. As a result ofthis, the higher reliability phase (crack initiation period) isvery short and crack propagation start at early stage of life.


×10532.521.510.5

Number of usage cycles, n

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Rel

iabi

lity,R

Plot-A: Reliability curve for successive

stress levels of 435, 360, and 320 MPa

Plot-B: Reliability curve for reverse sequence

of stress levels of 320, 360, and 435 MPa

Figure 9: Reliability at multi-stress levels of 435, 360 and 320 MPa.

But as life progresses the rate of crack propagation is slightlyreduced as compared to earlier rate at higher stress. In otherwords, when product is subjected to high-low sequence ofstress, the rate of loss of reliability is higher and starts atearly stage of life but slows down later when subjected to lowstress levels. This happens because of shorter crack initiationperiod due to high stress levels initially and relatively longercrack propagation periods when subjected to lower stresses.

4.3. Reliability Prediction for Multi-Stress Levels by ChangingSequence of Loading. To study the effect of change in se-quence of loading on reliability behavior of the helical com-pression spring, the sequence of multi-stress levels loading isreversed (i.e., 320 MPa, 360 MPa, and 435 MPa for successiveusage cycles 167750, 60000, and 40000, resp.).

The reliability plot for this reversed multilevel stressloading is plotted as plot-B in Figure 9. In multi-stressloading scenario, reversal of loading sequence will not affectthe fatigue life of spring much, as the fraction of lifeconsumed at different stress level remains the same in lightof linear damage accumulation theory. But reliability plot-Bindicates considerable change in its shape. The initial higherreliability phase is considerably longer due to low-high stresssequence. This is an indication of longer crack initiationphase and slower rate of damage accumulation because ofslow propagation of crack at lower stress levels. However, assoon as spring is subjected to higher stress level, the reliabilitystarts declining sharply due to faster propagation of cracksand hence higher rate of damage accumulation at higherstress level. In other words, the implications of change insequence of loading can be observed in the duration of crackinitiation phase and the rate of loss of reliability with usagetime (as shown in Figure 9). It can be observed that the rateof loss of reliability is generally higher for high-low sequenceof loading compared to low-high sequence but it changeslater.

Authors believe these two scenarios represent twoextremes but ideal cases of loading phenomenon of mul-tilevel stress loading. In real life applications, the loadingpattern is not necessarily sequential but could be any combi-nation of stress levels in any sequence of loading. Therefore,these two extreme scenarios provide a band of reliabilitybehavior for multilevel stress loading, and actual reliabilitybehavior, for any given real life loading pattern, could besomewhere within this band.

5. Conclusion

The proposed approach provides an easy methodologyfor modeling the probabilistic distribution of the damageaccumulation measure and hence capturing real life behaviorof the product. It proposes a simple and unique way tomodel the damage accumulation process treating it as a non-stationary process to capture damage accumulation and itsvariability at any given point of time. The proposed approachcan be extended to consider other fatigue failure distribu-tions such as lognormal and Weibull distributions to modelprobabilistic damage accumulation.

The major limitation of the proposed approach is that ittreats damage accumulation as linear phenomenon whereasin actual practice damage accumulation could be a nonlinearphenomenon especially in multi-stress loading scenarios.Our future research efforts are directed towards develop-ing an understanding to capture nonlinear damage accu-mulation phenomenon in multi-stress loading conditions.Another potential research area for future research can betowards capturing and incorporating degradation behavior(dynamic degradation phenomenon) into design optimiza-tion models to capture life-cycle issues and provide morerealistic solutions.

References

[1] K. Sobczyk, “Stochastic models for fatigue damage of materi-als,” Advances in Applied Probability, vol. 19, no. 3, pp. 652–473, 1987.

[2] Y. Liu and S. Mahadevan, “Stochastic fatigue damage model-ing under variable amplitude loading,” International Journal ofFatigue, vol. 29, no. 6, pp. 1149–1161, 2007.

[3] H. Shen, J. Lin, and E. Mu, “Probabilistic model on stochasticfatigue damage,” International Journal of Fatigue, vol. 22, no.7, pp. 569–572, 2000.

[4] P. H. Wirsching and Y. N. Chen, “Considerations ofprobability-based fatigue design for marine structures,” inMarine Structural Reliability Symposium, pp. 31–42, 1987.

[5] P. Albrecht, “S-N fatigue reliability analysis of highwaybridges, probabilistic fracture mechanics and fatigue methods:applications for structural design and maintenance,” AmericanSociety for Testing and Materials, vol. 789, pp. 184–204, 1983.

[6] W. F. Wu, H. Y. Liou, and H. C. Tse, “Estimation of fa-tigue damage and fatigue life of components under randomloading,” International Journal of Pressure Vessels and Piping,vol. 72, no. 3, pp. 243–249, 1997.

[7] M. A. Zaccone, “Failure analysis of Helical springs underCompressor Start/Stop Conditions,” ASM International, vol.1, no. 3, pp. 51–62, 2001.


[8] W. H. Munse, T. W. Wilbur, M. L. Tellalian, K. Nicoll, and K.Wilson, “Fatigue Characterization of fabricated ship details fordesign,” Ship Structure Committee, 1983.

[9] P. H. Wirsching, “Probabilistic fatigue analysis,” in Proba-bilistic Structural Mechanics Handbook, C. Sundararajan, Ed.,Chapman and Hall, New York, NY, USA, 1995.

[10] M. Nagode and M. Fajdiga, “On a new method for predictionof the scatter of loading spectra,” International Journal ofFatigue, vol. 20, no. 4, pp. 271–277, 1998.

[11] M. Liao, X. Xu, and Q. Yang, “Cumulative fatigue damagedynamic interference statistical model,” International Journalof Fatigue, vol. 17, no. 8, pp. 559–566, 1995.

[12] W.-F. Wu and T.-H. Huang, “Prediction of fatigue damage andfatigue life under random loading,” International Journal ofPressure Vessels and Piping, vol. 53, no. 2, pp. 273–298, 1993.

[13] M. Ben-Amoz, “A cumulative damage theory for fatigue lifeprediction,” Engineering Fracture Mechanics, vol. 37, no. 2, pp.341–347, 1990.

[14] E. Castillo, A. Fernandez-Canteli, and M. L. Ruiz-Ripoll, “Ageneral model for fatigue damage due to any stress history,”International Journal of Fatigue, vol. 30, no. 1, pp. 150–164,2008.

[15] J. Sethuraman and T. R. Young, “Cumulative damage thresh-old crossing models,” in Reliability and Quality Control,A. P. Basu, Ed., pp. 309–319, Elsevier, Amsterdam, TheNetherlands, 1986.

[16] A. Fatemi and L. Yang, “Cumulative fatigue damage andlife prediction theories: a survey of the state of the art forhomogeneous materials,” International Journal of Fatigue, vol.20, no. 1, pp. 9–34, 1998.

[17] P. Wang and D. W. Coit, “Reliability and degradation model-ing with random or uncertain failure threshold,” in Reliabilityand Maintainability Symposium, pp. 392–397, 2007.

[18] J. R. Benjamin and C. A. Cornell, Probability, Statistics, andDecision for Civil Engineers, McGraw-Hill, New York, NY,USA, 1970.

[19] W. Hwang and K. S. Han, “Cumulative damage modelsand multi-stress fatigue life prediction,” Journal of CompositeMaterials, vol. 20, no. 2, pp. 125–153, 1986.

[20] D. W. Coit, J. L. Evans, N. T. Vogt, and J. R. Thompson, “Amethod for correlating field life degradation with reliabilityprediction for electronic modules,” Quality and ReliabilityEngineering International, vol. 21, no. 7, pp. 715–726, 2005.

[21] F. G. Pascual and W. Q. Meeker, “Estimating fatigue curveswith the random fatigue-limit model,” Technometrics, vol. 41,no. 4, pp. 277–290, 1999.

[22] K. C. Kapur and L. R. Lamberson, Reliability in EngineeringDesign, John Wiley & Sons, New York, NY, USA, 1977.

[23] S. S. Rao, Reliability Based Design, McGraw-Hill, New York,NY, USA, 1992.

[24] C. S. Place, J. E. Strutt, K. Allsopp, P. E. Irving, and C. Trille,“Reliability prediction of helicopter transmission systemsusing stress-strength interference with underlying damageaccumulation,” Quality and Reliability Engineering Interna-tional, vol. 15, no. 2, pp. 69–78, 1999.

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